# Unified representation of the family of L-functions

## Abstract

The aim of this paper is to unify the family of L-functions. By using the generating functions of the Bernoulli, Euler and Genocchi polynomials, we construct unification of the L-functions. We also derive new identities related to these functions. We also investigate fundamental properties of these functions.

AMS Subject Classification:11B68, 11S40, 11S80, 26C05, 30B40.

## 1 Introduction

The theory of the family of L-functions has become a very important part in the analytic number theory. In this paper, using a new type generating function of the family of special numbers and polynomials, we construct unification of the L-functions.

Throughout this presentation, we use the following standard notions $N={1,2,…}$, $N 0 ={0,1,2,…}=N∪{0}$, $Z + ={1,2,3,…}$, $Z − ={−1,−2,…}$. Also, as usual denotes the set of integers, denotes the set of real number and denotes the set of complex numbers. We assume that $ln(z)$ denotes the principal branch of the multi-valued function $ln(z)$ with the imaginary part $ℑ(ln(z))$ constrained by $−π<ℑ(ln(z))≤π$.

Recently, the first author  introduced and investigated the following generating functions which give a unification of the Bernoulli polynomials, Euler polynomials and Genocchi polynomials:

$g a , b (x;t,k,β):= 2 1 − k t k e t x β b e t − a b = ∑ n = 0 ∞ Y n , β (x;k,a,b) t n n ! ,$
(1)

where ($|t|<2π$ when $β=a$; $|t|<|blog( β a )|$ when $β≠a$; $k∈ N 0$; $β∈C$ ($|β|<1$); $a,b∈C∖{0}$).

For the special values of a, b, k, b and β, the polynomials $Y n , β (x;k,a,b)$ provide us with a generalization and unification of the classical Bernoulli polynomials, Euler polynomials and Genocchi polynomials and also of the Apostol-type (Apostol-Bernoulli, Apostol-Euler, Apostol-Genocchi) polynomials.

Remark 1.1 If we set $k=a=b=1$ in (1), we get a special case of the generalized Bernoulli polynomials $Y n , β (x,k,1,1)$, that is, the so-called Apostol-Bernoulli polynomials $B n (x,β)$ generated by

$t β e t − 1 e x t = ∑ n = 0 ∞ B n (x,β) t n n !$

(cf. ).

Remark 1.2 By substituting $k+1=−a=b=1$ in (1), we are led to Apostol-Euler polynomials $E n (x,β)$ which are defined by means of the following generating function:

$2 β e t + 1 e x t = ∑ n = 0 ∞ E n (x,β)$

(cf. ).

Remark 1.3 Setting $k=−a=b=1$ into (1), we get the Apostol-Genocchi polynomials $G n (x,β)$ which are defined by means of the following generating function:

$2 t β e t + 1 e x t = ∑ n = 0 ∞ G n (x,β) t n n !$

(cf. ).

In terms of a Dirichlet character χ of conductor $f∈N$, Ozden et al.  extended and investigated the generating functions of the generalized Bernoulli, Euler and Genocchi numbers and the generalized Bernoulli, Euler and Genocchi polynomials with parameters a, b, β and k. Such χ-extended polynomials and χ-extended numbers are useful in many areas of mathematics and mathematical physics.

Definition 1.4 (Ozden et al. [, p.2783])

Let χ be a Dirichlet character of conductor $f∈N$. Then the aforementioned χ-extended generalized Bernoulli-Euler-Genocchi numbers $Y n , χ , β (k,a,b)$ and the aforementioned χ-extended generalized Bernoulli-Euler-Genocchi polynomials $Y n , χ , β (x;k,a,b)$ are given by the following generating functions:

$F χ , β (t;k,a,b)= 2 1 − k t k ∑ j = 1 f χ ( j ) ( β a ) b j e j t β b f e f t − a b f = ∑ n = 0 ∞ Y n , χ , β (k,a,b) t n n ! ,$
(2)

where ($|t|<2π$ when $β=a$; $|t|<|blog( β a )|$ when $β≠a$; $k∈ N 0$; $β∈C$ ($|β|<1$); $a,b∈C∖{0}$) and

$H χ , β (x,t;k,a,b)= F χ , β (t,k;a,b) e t x = ∑ n = 0 ∞ Y n , χ , β (x;k,a,b) t n n !$
(3)

($|t|<2π$ when $β=a$; $|t|<|blog( β a )|$ when $β≠a$; $k∈ N 0$; $β∈C$ ($|β|<1$); $a,b∈C∖{0}$).

Remark 1.5 Substituting $k=a=b=β=1$ into (2), we are led immediately to the generating function of the generalized Bernoulli numbers which are defined by means of the following generating function:

$∑ j = 1 f χ ( j ) t e j t e f t − 1 = ∑ n = 0 ∞ B n , χ t n n !$
(4)

(cf. ).

## 2 Unification of the L-functions

Our aim in this section is to apply the Mellin transformation to the generating function (3) of the polynomials $Y n , χ , β (x;k,a,b)$ in order to construct a unification of the various members of the family of the L-functions and to thereby interpolate $Y n , χ , β (x;k,a,b)$ for negative integer values of n.

Throughout this section, we assume that $β∈C$ with $|β|<1$ and $s∈C$.

By substituting (1) into (2), we obtain the following functional equation:

$F χ , β (t;k,a,b)= 1 f k ∑ j = 1 f χ(j) ( β a ) b j g a f , b ( j f , t f ; k , β f ) .$
(5)

By using this functional equation, we arrive at the following theorem.

Theorem 2.1 Let χ be a Dirichlet character of conductor f. Then we have

$Y n , χ , β (k,a,b)= f n − k ∑ j = 1 f χ(j) ( β a ) b j Y n , β f ( j f ; k , a f , b ) .$
(6)

By using (5), we modify (3) as follows:

$H χ , β (x,t;k,a,b)= 1 f k ∑ j = 1 f χ(j) ( β a ) b j g a f , b ( j + x f , t f ; k , β f ) .$
(7)

By using (7), we derive the following result.

Corollary 2.2 Let χ be a Dirichlet character of conductor $f∈N$. Then we have

$Y n , χ , β (x;k,a,b)= f n − k ∑ j = 1 f χ(j) ( β a ) b j Y n , β f ( j + x f ; k , a f , b ) .$
(8)

By applying the Mellin transformation to the generating function (1), Ozden et al. [, p.2784 Equation (4.1)] gave an integral representation of the unified zeta function $ζ β (s,x;k,a,b)$:

$ζ β (s,x;k,a,b)= 1 Γ ( s ) ∫ 0 ∞ t s − k − 1 g a , b (x;−t;k,β)dt ( min { ℜ ( s ) , ℜ ( x ) } > 0 ) ,$
(9)

where the additional constraint $ℜ(x)>0$ is required for the convergence of the infinite integral, which is given in (9), at its upper terminal. By making use of the above integral representation, Ozden et al. [, p.2784 Equation (4.1)] defined the unified zeta function $ζ β (s,x;k,a,b)$ as follows:

$ζ β (s,x;k,a,b)= ( − 1 2 ) k − 1 ∑ m = 0 ∞ β b m a b ( m + 1 ) ( m + x ) s ( β ∈ C ( | β | < 1 ) ; s ∈ C ( ℜ ( s ) > 1 ) ) .$
(10)

By applying the Mellin transformation to the generating function (7), we have the following integral representation of the unified two-variable L-functions $L χ , β (s,x;k,a,b)$: (11)

in terms of the generating function $H χ , β (x,t;k;a,b)$ defined in (7). By substituting (9) into ( 11), we obtain

$L χ , β (s,x;k,a,b)= 1 f k + s ∑ j = 1 f χ(j) ( β a ) b j ζ β f ( s , j + x f ; k , a f , b )$
(12)

where ($β∈C$ ($|β|<1$); $s∈C$ ($ℜ(s)>1$)).

Consequently, by making use of (10) and (12), we are ready to define a two-variable unification of the Dirichlet-type L-functions $L χ , β (s,x;k,a,b)$ as follows.

Definition 2.3 Let χ be a Dirichlet character of conductor $f∈N$. For $s,β∈C$ ($|β|<1$), we define a two-variable unified L-function $L χ , β (s,x;k,a,b)$ by

$L χ , β (s,x;k,a,b)= f − k ( − 1 2 ) k − 1 ∑ m = 0 ∞ β b m χ ( m ) a b ( m + f ) ( m + x ) s ( β ∈ C ( | β | < 1 ) ; ℜ ( s ) > 1 ) .$
(13)

Remark 2.4 If we substitute $x=1$ into (13), we get the unified L-function

$L χ , β (s;k,a,b):= L χ , β (s,1;k,a,b)$

by

$L χ , β (s;k,a,b)= f − k ( − 1 2 ) k − 1 ∑ m = 1 ∞ β b m χ ( m ) a b ( m + f ) m s ,$

where ($ℜ(s)>1$, $β∈C$ ($|β|<1$)).

Remark 2.5 Upon substituting $k=a=b=1$ and $β= ξ u$ into (13), we arrive at the interpolation function for twisted generalized Eulerian numbers and polynomials, which is given as follows:

$l 1 ( u ξ , s , χ ) = L χ , ξ u (s,x;1,1,1),$

where, for a positive integer r, ξ is the r th root of 1.

$l 1 ( u ξ , s ; χ ) = ∑ m = 0 ∞ ( ξ u ) m χ ( m ) ( m + x ) s$

(cf. ).

Remark 2.6 Substituting $x=1$ into (13), we get a unification of the L-functions

$L χ , β (s,1;k,a,b)= L χ , β (s;k,a,b).$

Substituting $χ≡1$ into (13), we get a unification $ζ β (s,x;k,a,b)$ of the Hurwitz-type zeta function which is given in (10). We also note that both the Hurwitz (or generalized) zeta function

$ζ(s,x)= ζ 1 (s,x;1,1,1)= ∑ n = 0 ∞ 1 ( n + x ) s$

(cf. [27, 28]) and the Riemann zeta function

$ζ(s)= ζ 1 (s,1;1,1,1)= ∑ n = 1 ∞ 1 n s$

are obvious special cases of the unified zeta function $ζ β (s,x;k,a,b)$ (cf. [16, 27, 28]). The relationship between the unified zeta function and the Hurwitz-Lerch zeta function $Φ(z,s,a)$ was given by Ozden et al. :

$ζ β (s,x;k,a,b):= ( − 1 2 ) k − 1 a − b Φ ( β b a b , s , x ) ,$
(14)

where the Hurwitz-Lerch zeta function is defined by

$Φ(z,s,x)= ∑ n = 0 ∞ z n ( n + x ) s ,$

which converges for ($x∈ C ╲ Z 0 −$, $s∈C$ when $|z|<1$; $ℜ(s)>1$ when $|z|=1$), where as usual

$Z 0 − = Z − ∪{0}$

(cf. [27, 28]).

A relationship between the functions $L χ , β (s,x;k,a,b)$ and $ζ β (s,x;k,a,b)$ is provided by the next theorem.

Theorem 2.7 Let $s∈C$. Let χ be a Dirichlet character of conductor $f∈N$. Then we have

$L χ , β (s,x;k,a,b)= f − s − k ∑ j = 1 f ( β a ) j b χ(j) ζ β f ( s , j + x f ; k , a f , b ) .$
(15)

Proof Substituting $m=nf+j$, $j=1,2,…,f$, $n=0,…,∞$ into (13), we obtain

$L χ , β (s,x;k,a,b)= ( − 1 2 ) k − 1 f − s − k ∑ j = 1 f ( β a ) j b χ(j) ∑ n = 0 ∞ β b n f a b n f ( n + j + x f ) s .$

After some algebraic manipulations, we arrive at the desired result. □

Remark 2.8 Substituting $a=b=k=1$ into (13), we have

$L χ , β (s,x;1,1,1)= ∑ m = 0 ∞ β m χ ( m ) ( m + x ) s ( ℜ ( s ) > 1 , β ∈ C ( | β | < 1 ) )$

which interpolates the Apostol-Bernoulli polynomials attached to the Dirichlet character, which are given by means of the following generating functions:

$∑ j = 1 f χ ( j ) t β j e t ( j + x ) β f e t f − 1 = ∑ n = 0 ∞ B n , χ (x,β) t n n ! .$

Let f be an odd integer. If we set $a=−1$ and $k=0$ into (13), then we have

$L χ , β (s,x;1,−1,1)=2 ∑ m = 1 ∞ ( − 1 ) m χ ( m ) β m ( m + x ) s ( ℜ ( s ) > 1 , β ∈ C ( | β | < 1 ) ) ,$

which interpolate the Apostol-Euler polynomials attached to the Dirichlet character, which are defined by the following generating functions:

$∑ j = 1 f 2 χ ( j ) β j e t ( j + x ) β f e t f + 1 = ∑ n = 0 ∞ E n , χ (x,β) t n n !$

(cf. ).

By using (15) and (14), we arrive at the following result.

Corollary 2.9 Let $s∈C$. Let χ be a Dirichlet character of conductor $f∈N$. Then we have

$L χ , β (s,x;k,a,b)= ( − 1 2 ) k − 1 a − f b f − s − k ∑ j = 1 f ( β a ) j b χ(j)Φ ( β f b a f b , s , j + x f ) .$

Theorem 2.10 Let χ be a Dirichlet character of conductor f. Let n be a positive integer. Then we have

$L χ , β (1−n,x;k,a,b)= ( − 1 ) k f ( n − 1 ) ! ( n + k − 1 ) ! Y n + k − 1 , χ , β (x;k,a,b).$
(16)

Proof By substituting $s=1−n$ into (15), we get

$L χ , β (1−n,x;k,a,b)= f n − 1 − k ∑ j = 1 f ( β a ) j b χ(j) ζ β f ( 1 − n , j + x f ; k , a f , b ) .$

By using Theorem 7 in , we get By substituting (8) into the above, we arrive at the desired result. □

Remark 2.11 The two-variable Dirichlet L-function and the Dirichlet L-function are obvious special cases of the unified Dirichlet-type L-functions $L χ , β (s,x;k,a,b)$ defined by (13). We thus have (cf. )

$L(s,x;χ)= ∑ m = 0 ∞ χ ( m ) ( m + x ) s$

and

$L(s;χ)= ∑ m = 1 ∞ χ ( m ) m s ,$

where $ℜ(s)>1$. By analytic continuation, this function can be extended to a meromorphic function on the whole complex plane. We have

$L(1−n;χ)=− B n , χ n ,$

where $n∈ Z +$ and $B n , χ$, the usual generalized Bernoulli number, is defined by (4). The Dirichlet L-function is used to prove the theorem on primes in arithmetic progressions. Dirichlet shows that $L(s;χ)$ is non-zero at $s=1$. Furthermore, if χ is a principal character, then the corresponding Dirichlet L-function has a simple pole at $s=1$ (cf. [6, 7, 9, 18, 24, 27, 28, 30, 31]).

## 3 Applications

In this section, by using (16) and the following formula, which was proved by Ozden et al. [, Theorem 5, Equation (3.10)]

$Y n , χ , β (x;k,a,b)= ∑ j = 0 n ( n j ) x n − j Y j , χ , β (k,a,b),$
(17)

we construct a meromorphic function involving a unified family of L-functions. Therefore, using (16) and (17),

$L χ , β (1−n,x;k,a,b)= x n + k − 1 f ∏ l = 0 k − 1 ( n + l ) ∑ j = 0 n + k − 1 ( n + k − 1 j ) 1 x j Y j + k − 1 , χ , β (k,a,b).$

From the above equation, we arrive at the following theorem.

Theorem 3.1 Let $x≠0$. Let χ be a Dirichlet character of conductor f. Then we have

$L χ , β (s,x;k,a,b)= x k − s f ∏ l = 0 k − 1 ( s − 1 − l ) ∑ j = 0 ∞ ( k − s j ) 1 x j Y j + k − 1 , χ , β (k,a,b).$

The function $L χ , β (s,x;k,a,b)$ is an analytic function at $s=0$. We now compute the value of this function at this point as follows:

$L χ , β (0,x;k,a,b)= x k ( − 1 ) k f ∏ l = 0 k − 1 ( 1 + l ) ∑ j = 0 k ( k j ) 1 x j Y j + k − 1 , χ , β (k,a,b).$

The function $L χ , β (s,x;k,a,b)$ is a meromorphic function. This function has simple poles which are

$s=1,2,3,…,k.$

The residues of this function at the simple poles at $s=1$ and $s=k$ are given, respectively, as follows:

$Res s = 1 { L χ , β ( s , x ; k , a , b ) } = x k − 1 f ( − 1 ) k ∏ l = 0 k − 1 ( 2 + l ) ∑ j = 0 k − 1 ( k − 1 j ) 1 x j Y j + k − 1 , χ , β (k,a,b)$

and

$Res s = k { L χ , β ( s , x ; k , a , b ) } = Y k − 1 , χ , β ( k , a , b ) f ∏ l = 0 k − 2 ( k − 1 − l ) .$

Remark 3.2 Simsek (cf. [20, 21]) defined a twisted two-variable L-function $L ξ , q ( h ) (s,x;χ)$ as follows:

$L ξ , q ( h ) (s,x;χ)= ∑ m = 0 ∞ χ ( m ) ϕ ξ ( m ) q h m ( x + m ) s − log q h s − 1 ∑ m = 0 ∞ χ ( m ) ϕ ξ ( m ) q h m ( x + m ) s − 1 ,$

where $q∈C$ ($|q|<1$); $ξ r =1$ ($r∈Z$); $ξ≠1$. Observe that if $ξ=1$, then $L ξ , q ( h ) (s,x;χ)$ is reduced to the work of Kim .

Relationship between the function $L χ , β (s,x;k,a,b)$ and $L ξ , q ( h ) (s,x;χ)$ is given as the following result.

Corollary 3.3 Let χ be a Dirichlet character of conductor f. Then we have

$L 1 , β b a b ( b ) (s,x;χ)= ( − 2 ) k a b f f k ( L χ , β ( s , x ; k , a , b ) − log q h s − 1 L χ , β ( s − 1 , x ; k , a , b ) ) .$

We conclude our present investigation by remarking that the existing literature contains several interesting generalizations and extensions of the Hurwitz-Lerch zeta function $Φ(z,s,a)$, Hurwitz zeta function $ζ(s,x)$ and L-function (cf. ); see also the references cited in each of these earlier works.

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## Acknowledgements

Dedicated to Professor Hari M Srivastava.

Both authors are partially supported by Research Project Offices Akdeniz Universities and the Commission of Scientific Research Projects of Uludag University Project number UAP(F) 2011/38 and 2012/16. We would like to thank referees for their valuable comments.

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Correspondence to Hacer Ozden.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

Both authors completed the paper together. Both authors read and approved the final manuscript.

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Ozden, H., Simsek, Y. Unified representation of the family of L-functions. J Inequal Appl 2013, 64 (2013). https://doi.org/10.1186/1029-242X-2013-64

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### Keywords

• Bernoulli numbers
• Bernoulli polynomials
• Euler numbers
• Euler polynomials
• Genocchi numbers
• Genocchi polynomials
• Dirichlet L-functions
• Hurwitz zeta function
• Riemann zeta function 